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IIUM Engineering Journal, Vol. 1, No. 1, January 2000 M. H. Ali et al.
7
STRESS ANALYSIS OF POLARIZATION MAINTAINING
OPTICAL FIBERS BY THE FINITE ELEMENT METHOD
M. H. Aly
Faculty of Engineering, University of Alexandria, Alexandria 21544, Egypt,
A. S. Farahat, M. S. Helmi and M. Farhoud
Faculty of Science, University of Alexandria, Alexandria, Egypt
Abstract: Stress-induced birefringence in single mode
polarization maintaining optical fibers has been
investigated using the finite element method. The modal
birefringence caused by external forces in the Panda and
the Side Tunnel fibers are calculated. It is found that the
modal birefringence is directly proportional to the radial
distance from the fiber center. As expected, the modal
birefringence vanishes with the variation in the
magnitude of the applied external loads.
Key Words: Birefringence, Polarization, Panda Fiber,
Side-Pit Fiber, Finite Element Method.
1. INTRODUCTION
Polarization maintaining (PM) optical fibers are of great
importance in optical coherent communications and
some optical fiber sensor applications[1]. The fibers that
preserve the state of polarization of the guided light can
be achieved by destroying the circular symmetry of the
refractive index distribution in the core region.
However, the birefringence properties of this kind of
fiber are much more limited when compared with what
can be achieved with stress induced birefringence.
Recently, a variety of structures has been proposed for
the polarization maintaining fibers with stress induced
birefringence, most of which are designed to maintain a
differential stress in the fiber core region[1]. There are
many such successful fiber structure designs, such as
Panda, Bow Tie, Side Tunnel and Side Pit fibers[1].
Shih[2] described a new fiber manufacturing technique
called preform deformation. Using this technique, a new
kind of PM fiber with a circular core, an elliptical stress
applying cladding, and an elliptical jacket can be made.
This has been studied experimentally by D. Marcuse et
al.[3]. In our work, full advantage of the finite element
method is taken to develop a numerical study of the
birefringence due to the effects of external stresses in
both Side Pit and the Side Tunnel fibers. The
birefringence PM fibers manufactured through preform
deformation are analyzed. From the calculated stress
distribution in the fiber, the refractive index tensor
change is obtained.
The finite element method is an approximate
treatment of physical problems, defined, for example,
by differential equations with boundary conditions, or
by variation principles. Compared to other approximate
methods it is successful when the domain of the
problem has a complicated shape or when the function
involved behaves differently in different parts of the
domain. The domain is represented approximately by a
collection of finite numbers of connected subdomains of
simple shape (e.g., triangles), called finite elements.
The finite element method will to be applied in this
paper for the determination of the stress distribution and
the modal birefringence, due to the effects of external
forces acting on the fiber cross section, for the Panda
and the Side Tunnel type fibers. The fiber cross section
is first divided into “elements”. In this research, a
triangular element is used. A linear system of equations
can be established according to the principles of the
finite element method, and then the stress distribution
and, further, the refractive index tensor distribution may
be calculated in each element. The details of the use of
the finite element method can be found in the
literature[4-5].
2. FORMULATION OF THE PROBLEM BY
THE FINITE ELEMENT METHOD
2.1 Matrix Solution Techniques
Solving the waveguide problem by the finite element
method, the key factor affecting storage requirements
and computational efforts is the choice of the algorithm
to solve the matrix equation. The advantage of higher-
order basis functions for the fields is that they give a
more accurate solution. But it involves an increased
programming effort, particularly when considering
anisotropic materials, in the finite element, and penalty
functions[6]. Another advantage of the higher order
polynomials (for given matrix order) is that it increases
the density of the matrix. It should be noted the tradeoff
optimum choice between low and high order
polynomials depends on the matrix algorithm used.
2.2 Principle of Minimum Potential Energy
The basic principle of the finite element method is that a
continuum (the total structure) can be modeled
analytically by its subdivision into regions.
The principle of minimum potential energy can be
used following Ref. [1]. Among all displacements of
admissible form, those that satisfy the equilibrium
IIUM Engineering Journal, Vol. 1, No. 1, January 2000 M. H. Ali et al.
8
conditions make the potential energy assume a
stationary (minimum) value. The total potential energy
of a structure can be expressed as a strain energy, U,
plus the potential energy, V, of the applied loads, i.e.,
U V (1)
The strain energy U of a complete structure consisting
of N elements is simply the sum of the N element strain
energies
e
T
teeee
N
1e
e
N
1e
fuuKu
2
1
UU , (2)
where the superscript t is the operator to traznspose the
vector or the matrix, and {ue}, [Ke], and {fTe} are the
vector of nodal points displacements, the stiffness
matrix, and the initial force vector in the eth element,
respectively. Equation (2) is rearranged to construct the
global equation as:
U u K u u ft t T
1
2
, (3)
where {u}, [K], and {fT} denote the global nodal point
displacement vector, the global stiffness matrix and the
global initial force vector, respectively.
The potential energy, V, due to external load is
expressed by:
V u ft L , (4)
where {fL} is the global load vector acting on each
nodal point. Inserting expressions for U and V from Eq.
(3) and Eq. (4) into Eq. (1), one can get:
1
2
u K u u f u ft t T
t
L , (5)
Applying to this the necessary condition for minimum
energy, i.e.,
u
K u f fT L 0 , (6)
K u f fT L (7)
This equation gives the displacement at all nodal points
of the fiber under external forces. Once the global nodal
point displacement vector, {u}, is determined, stress in
each element is given by[1]:
e e e e oeD B u . (8)
with e = 1,2,3...N, where [De], [Be], and{o} are the
material stiffness matrix, the matrix relating nodal
displacements to strain field, and the initial strain vector
in the eth element, respectively.
3. DERIVATION OF STRAIN ENERGY
As in the case of optical fibers, when the dimension of
the structure in one direction (the z-direction) is very
large in comparison with the other two transverse
directions (x- and y-) and the applied forces act in the x-
y plane and do not vary in the z-direction, the problem
becomes a “plane strain problem”[1].
In the plane strain problem, the longitudinal strain z
is zero, as are the shear strains �yz and �zx. Introducing
the condition on z into the relevant strain-stress
equation we get:
21
1
211
TEE
yxx
(9)
21
1
211
TEE
yxy
(10)
xyxy
E
12
, (11)
where x, y and x, y denote the normal stresses and
strains, respectively, and xy, and xy are the shear stress
and strain, respectively, E is the elastic modulus, is
Poisson’s ratio, is the thermal expansion coefficient,
and T is the temperature change (negative on cooling).
These equations are expressed in the form:
oD , (12)
where {}and {} denote the stress and strain vectors,
respectively, defined as:
x
y
xy
, (13)
and
x
y
xy
, (14)
0 denotes the initial strain, where:
o 1
1
1
0
T , (15)
and [D] is the material stiffness matrix given by[1]:
D
E
1 1 2
1 0
1 0
0 0 1 2 2
, , ,
, , ,
, ,
. (16)
The strain energy in plain strain is given by[1]:
U dxdy
S
t
o
L
2
, (17)
where L is the length of the optical fiber and the integral
is carried out over any region S under consideration. In
IIUM Engineering Journal, Vol. 1, No. 1, January 2000 M. H. Ali et al.
9
this paper a two-dimensional triangular element is used
to determine the variation of stresses in optical fibers.
The formulation of this element is well developed and
described in many references [6]. The eth element
stiffness matrix and initial force vector are:
ee
te
BDB S L
e
K , (18)
eoete DB S L
e
Tf , (19)
L is the fiber length and [ke] and {fT
e} denote the
element stiffness matrix and the element initial force
vector, respectively.
4. ACCURACY OF THE FINITE ELEMENT
ANALYSIS
Before conducting the stress analysis of noncircular
core fibers, the accuracy of the finite element analysis
itself is first investigated.
We can estimate the amount of error by comparing
the finite element solution with the analytical solutions
that show the same behavior.
1. Thermal stress: This is the problem of thermal
stress developing in long concentric cylinders of
different materials, which are joined together at some
initial temperature and their temperature is then
changed. This has been treated by B.M. Azizur et al[7].
Stress distributions in the cylinder are given as:
r
b a
b
E
2 2
2
2 1
2 1
T , (0 r a) (20)
=
a
2b
T
b
r
2
2
2
2
2 1
1
1E , (0 r b) (21)
b a
b
E
2 2
2
2 1
2 1
T , (0 r a) (22)
=
a
2b
T
b
r
2
2
2
2
2 1
1
1E , (0 r b) (23)
where a and b are respectively the inner and outer radii
of the concentric cylinder, E, , and T are elastic
modules, Poisson’s ratio and temperature change in the
medium, and 1 and 2 are the thermal expansion
coefficients of the mediums 1 and 2, respectively.
2- Stress by loads: Stress distributions on the x-axis
induced by the force Wo acting on the cylinder along the
vertical y- axis direction of the cylinder are expressed
as:
x
oW b x
b x
b
1
4 2 2
2 2 2
, (24)
b
y
oW b
b x
4
1
4
2 2 2
, (25)
3- Stress fields by finite element analysis:
One-quarter of the entire cross section is covered by
elements, since the optical fiber under consideration has
two symmetry lines, one along the x-axis and the other
along the y-axis. Typically, we used 140 elements and
93 nodal points in this section. The error in the results
obtained by the finite element analysis is less than 4
percent of the analytical results[1].
5. RESULTS AND DISCUSSION
A computer program coded in MATLAB is used for the
analysis of the birefringence properties of optical fibers
using the finite element method.
The structure of the Side Tunnel and Panda fibers is
shown in Figs. 1 and 2. The fiber parameters used in
this study are shown in Table 1.
Taking into consideration that, in the Side Tunnel
fiber, d is the distance from the nearest edge of SAZ to
the core center and all the refractive indices n1, n2, and
n3 have no effect on the strain.
Fig. 1 Side Tunnel fiber
Fig. 2 Panda fiber
Figure 3 shows the stress distribution x and y on
the x-axis of the Panda type fiber when force is applied
along the y-axis. It can be observed from the curve that
stresses
Table I Parameters of the Panda and the Side Tunnel fibers
y
a
x
n
d2
b
d1
y
n2 n1
a
x
n3
c
n3
b
d
IIUM Engineering Journal, Vol. 1, No. 1, January 2000 M. H. Ali et al.
10
Side Tunnel
fiber
Panda fiber
2a (m) 5 ~ 10 10
2b (m) 125 120
1 (o C -1)
2 (o C -1)
3 (o C -1)
7.6 10 -6
5.4 10 -7
3.66 10 -3
2.125 10 -6
5.4 10 -7
1.45 10 -6
d/c
a/d
0.15
2.0 ~ 4.0
-
-
E (kg /mm2) 7830 7830
0.186 0.186
2c (m) - 30
d1 (m) - 20
d2 (m) - 35
are nearly constant in the core region of the fiber. For
the clad region between the core region and the SAZ
region, the first clad region, stresses decrease gradually.
Reaching the SAZ, it is observed that the stresses nearly
equal zero. The behavior of the stresses in the core and
the first clad region can be explained by the resultant
stresses because external and internal stresses existing at
these regions are not equal and the net stress varies
uniformly as we proceed along the fiber cross section.
After this region, the stresses are nearly equal resulting
in a nearly zero stress.
-12
-10
-8
-6
-4
-2
0
2
4
1.14 4.57 8.34 14.29 24 42.29
x( m)
S
tr
es
s
(k
g/
m
m2
)
y
x
Fig. 3 Stress distribution on the x-
axis when force is applied along y-
axis in the Panda fiber.
Figure 4 shows the stress distribution on the y-axis
when force is applied along the x-axis in the Panda
fiber. One can observe the same behavior as in Fig. 3.
As we reach the clad region, one observes that stresses
decrease gradually. The decrement of the stresses along
the y-axis can be explained by the decrement of the
external stresses because the area is affected by the
external normal force along the axis that increases when
proceeding along the axis. The applied load, Wo, in both
Figs. 3 and 4 is 0.5 kg/cm.
Based on Ref. [8] and stresses shown in Fig. 3, the
stress-induced modal birefringence in the Panda type
fiber when force is applied along the x-axis direction is
shown in Fig. 5. It is observed from the figure that the
modal birefringence is directly proportional to the radial
distance from the fiber center. This is due to the effect of
the stresses at this region. The value of the
birefringence also depends on the magnitude of the
external applied forces. Similarly, the stress-induced
modal birefringence when force is applied along the y-
axis is shown in Fig. 6. The figure shows nearly the
same behavior as in Fig. 5. It is clear that the
birefringence at this applied load equals zero at the core
region.
Changing the applied load, Wo, the stress difference,
x - y, is calculated when the force is applied in x- and
y- directions, respectively. From this difference, the
stress-induced modal birefringence is obtained.
-10
-8
-6
-4
-2
0
2
4
1.14 6.97 17.03 42.29
y( m )
S
tr
es
s
(
kg
/m
m
2 )
Fig. 4 Stress distribution on the y-
axis when force is applied along x-
axis in the Panda fiber.
y
x
0
5
10
15
20
25
1.
14
3.
43
5.
83
8.
34
11
.7 17 24
34
.3
49
.1
x (m)M
od
al
B
ire
fr
in
ge
nc
e,
B
x1
05
Fig. 5 Modal birefringence in the
Panda fiber when force is applied
along x-axis.
IIUM Engineering Journal, Vol. 1, No. 1, January 2000 M. H. Ali et al.
11
-2
0
2
4
6
8
10
12
14
16
18
1.14 5.83 11.66 24 49.14
y(m)
Fig. 6 Modal birefringence in the
Panda fiber when force is applied
along y-axis direction.
M
od
al
B
ire
fr
in
ge
nc
e,
B
x
1
05
The stress difference in the Panda fiber when force
is applied along x-axis is shown in Fig. 7, from which
it is noted that the stress difference decreases with the
applied loads on the fiber. This means that the stress
along the y-axis is greater than the stress along the x-
axis. Figure 8 shows the modal birefringence in the
Panda fiber when force is applied along the x-axis. It
is clear that the modal birefringence decreases with the
applied loads on the fiber.
-16
-14
-12
-10
-8
-6
-4
-2
0
0.2 0.4 0.6 0.8 1 1.2 1.4
Wo (kg/cm)
S
tr
es
s
di
ffe
re
nc
e
x-
y
(k
g/
m
m
2 )
Fig. 7 Stress difference in the Panda fiber when force is
applied along the x-axis.
-6
-5
-4
-3
-2
-1
0
0.2 0.4 0.6 0.8 1 1.2 1.4
Wo (kg/cm)
M
od
al
b
ire
fr
in
ge
nc
,
B
(x
1
0
5 )
Fig. 8 Modal birefringence in the Panda fiber
when force is applied along x-axis.
0
5
10
15
20
25
30
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Wo (kg/cm)
S
tr
es
s
di
ffe
re
nc
e,
x-
y
(k
g/
m
m
2 )
Fig. 9 Stress distribution in the Panda fiber
when force is applied along y-axis.
Figure 9 shows the stress distribution in the Panda
fiber when force is applied along the y-axis. A direct
proportionality between the applied loads, Wo, and the
stress difference is noticed. Figure 10 shows the modal
birefringence in the Panda fiber when force is applied
along the y-axis, where an increase is noticed in the
modal birefringence with the applied loads. From these
results it is expected that we can control the magnitude
of the birefringence in the fiber by controlling the
external loads acting on the fiber.
Figures 11 and 12 show the stress distribution for the
Side Tunnel fiber on the x-axis and the y-axis
directions. The two figures show nearly the same
behavior and the same order of magnitude as the Panda
fiber. This is expected since the two fibers are different
only in the position of the SAZs and the parameters
This is repeated for the modal birefringence, Fig. 13 and
14. As the force is applied along the y-direction in the
fiber, one observes that the modal birefringence
vanishes nearly at the limit of the fiber. It is also
expected that the modal birefringence will vanish as the
magnitude of the external applied loads, oW , is varied.
Similar to the Panda fiber, the behavior of the Side
Tunnel fiber is shown in Figs. 15 to 18.
Fig. 10 Modal birefringence in the Panda
fiber when force is applied along y-axis.
0
1
2
3
4
5
6
7
8
9
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Wo (kg/cm)
M
od
al
b
ire
fr
in
ge
nc
e,
B
(x
10
5 )
IIUM Engineering Journal, Vol. 1, No. 1, January 2000 M. H. Ali et al.
12
.
-12
-10
-8
-6
-4
-2
0
2
4
6
8
10
1.19 6.07 12.14 25 51.19x(m)
S
tr
es
s,
(
kg
/m
m
2 )
Fig.11 Stress distribution for the Side
Tunnel fiber on the x-axis when force is
applied along y-axis.
x
y
-12
-10
-8
-6
-4
-2
0
2
4
6
8
10
1.19 6.07 12.14 25 51.19
y(m)
S
tr
es
s,
(k
g/
m
m
2 )
Fig. 12 Stress distribution for the Side
Tunnel fiber on the y-axis when force is
applied along x-axis.
y
x
0
5
10
15
20
25
1.19 6.07 12.14 25 51.19
x(m)
M
od
al
B
ire
fr
in
ge
nc
e,
B
x1
05
Fig. 13 Modal birefringence in the Side
Tunnel fiber when force is applied along the
x-axis.
-5
0
5
10
15
20
25
1.19 6.07 12.14 25 51.19
y (m)
M
od
al
B
ire
fr
in
ge
nc
e,
B
x
1
05
Fig. 14 Modal birefringence in the Side
Tunnel fiber when force is applied along the
y-axis.
-60
-50
-40
-30
-20
-10
0
0.2 0.4 0.6 0.8 1 1.2 1.4
Wo (kg/cm)
S
tr
es
s
di
ffe
re
nc
e,
x-
y
(k
g/
m
m
2 )
Fig. 15 Stress difference in the Side Tunnel
fiber when force is applied along x-axis.
-2
-1.8
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2 0.4 0.6 0.8 1 1.2 1.4
W o (kg/cm)
M
od
al
B
ire
fr
in
ge
nc
e,
B
(x
10
5 )
Fig. 16 Modal birefringence in the Side
Tunnel fiber when force is applied along x-
axis.
IIUM Engineering Journal, Vol. 1, No. 1, January 2000 M. H. Ali et al.
13
0
5
10
15
20
25
30
35
40
0.2 0.4 0.6 0.8 1 1.2 1.4
Wo (kg/cm)
S
tr
es
s
di
ffe
re
nc
e
x-
y
(k
g/
m
m
2 )
Fig. 17 Stress difference in the Side Tunnel
fiber when force is applied along y-axis.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0.2 0.4 0.6 0.8 1 1.2 1.4
Wo (kg/cm)
M
od
al
b
ire
fr
in
ge
nc
e,
B
(x
10
5 )
Fig. 18 Modal birefringence in the Side
Tunnel fiber when force is applied along
the y-axis.
6. CONCLUSION
The finite element method has been applied for both the
Side Tunnel and Panda fibers to obtain variation in
stresses and modal birefringence.
It is found that the modal birefringence is directly
proportional to radial distance from the fiber center. It is
also found that as the force is applied, the modal
birefringence vanishes nearly at the core limits of the
fiber.
REFERENCES
[1] K. Okamoto, T. Hosaka and T. Edahiro, “Stress
Analysis of Optical Fibers by a Finite Element Method”,
IEEE J. Quantum Electron, Vol. QE-17, No. 10, pp.
2123-2129, 1981.
[2] S. C. Chao, “Extended Gaussian Approximation for
Single-Mode Graded Index Fibers”. IEEE J. Lightwave
Technol., Vol. LT-12 (3), pp. 392-395, 1994.
[3] D. Marcuse and C. Lin, “Low Dispersion Single-Mode
Fiber Transmission”, IEEE J. Quantum Electron, Vol.
QE-17, No. 6, pp. 869-877,1981.
[4] C. X. Sho and U R.Q. Hui, “Polarization Coupling in
Single-Mode Single-Polarization Fibers”, Opt. Lett.,
Vol. 13, No. 12, pp. 1120-1122, 1988.
[5] P. K. Bachmann, Dieter Leers, Hermann Wehr and Erich
R. Wehrhahn, “Dispersion-Flattened Single Mode Fibers
prepared with PCVD: Performance, Limitations, Design
Optimization”, IEEE J.Lightwave Technol., Vol. LT-4
(7), pp. 858-863, 1983.
[6] K. T. Bathe, Finite Element Procedures, Prentice Hall
Inc., 1996.
[7] B. M. A. Rahaman, A. Fernandez, and B. Davies,
“Review of Finite Element Methods for Microwave and
Optical Waveguides”, IEEE J. Lightwave Technol., Vol.
LT-13, pp. 1442-1446, 1991.
[8] K. Hayata and M. Koshiba, “Stress-induced
birefringence of Side-Tunnel Type Polarization-
Maintaining Fiber”, IEEE J. Lightwave Technol., Vol.
LT-4, 1986.
IIUM Engineering Journal, Vol. 1, No. 1, January 2000 M. H. Ali et al.
14
BIOGRAPHY
Prof. Dr. Moustafa Hussein Aly is currently Professor of
Engineering Physics, Faculty of Engineering, University
of Alexandria, Alexandria, Egypt. He was born in
Alexandria in 1953. He received his B.Sc. in 1976 in
Communications and Electrophysics, his M.Sc. in 1983
and his Ph.D. in 1987 in Engineering Physics, all from
Faculty of Engineering, University of Alexandria,
Egypt. He is a member of the Optical Society of
America (OSA) and of the Egyptian Society of Solid
State (ESSS). His area of interest is Laser and Fiber
Optics where he has about 40 publications. E-mail:
mosaly@hotmail.com
Mr. Ashraf M. S. Farahat is currently a lecturer,
Department of Physics, Faculty of Science, University
of Alexandria, Alexandria, Egypt. He was born in
Alexandria in 1972. He received his B.Sc. in 1992 and
M.Sc. in 1999, both in Physics, Faculty of Science,
University of Alexandria, Egypt.
Dr. Maher Farhoud is currently Associate Professor,
Department of Physics, Faculty of Science, University
of Alexandria, Alexandria, Egypt. He was born in
Alexandria in 30/11/1958. He received his B.Sc. in
1980 and M.Sc. in 1986, both in Physics, Faculty of
Science, University of Alexandria, Egypt. He received
his Ph.D. in Physics, Adam Mickiwicz University,
Poland. His area of interest is Laser and Nonlinear
Optics. E-mail: mfarhoud@yahoo.com.